121 research outputs found
Homotopy Canonicity for Cubical Type Theory
Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral
Strict Rezk completions of models of HoTT and homotopy canonicity
We give a new constructive proof of homotopy canonicity for homotopy type
theory (HoTT). Canonicity proofs typically involve gluing constructions over
the syntax of type theory. We instead use a gluing construction over a "strict
Rezk completion" of the syntax of HoTT. The strict Rezk completion is specified
and constructed in the topos of cartesian cubical sets. It completes a model of
HoTT to an equivalent model satisfying a saturation condition, providing an
equivalence between terms of identity types and cubical paths between terms.
This generalizes the ordinary Rezk completion of a 1-category
CANONICITY AND HOMOTOPY CANONICITY FOR CUBICAL TYPE THEORY
Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. We present in this article two canonicity results, both proved by a sconing argument: a homotopy canonicity result, every natural number is path equal to a numeral, even if we take away the equations defining the lifting operation on the type structure, and a canonicity result, which uses these equations in a crucial way. Both proofs are done internally in a presheaf model
Internal Parametricity for Cubical Type Theory
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity
Guarded Cubical Type Theory: Path Equality for Guarded Recursion
This paper improves the treatment of equality in guarded dependent type
theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an
extensional type theory with guarded recursive types, which are useful for
building models of program logics, and for programming and reasoning with
coinductive types. We wish to implement GDTT with decidable type-checking,
while still supporting non-trivial equality proofs that reason about the
extensions of guarded recursive constructions. CTT is a variation of
Martin-L\"of type theory in which the identity type is replaced by abstract
paths between terms. CTT provides a computational interpretation of functional
extensionality, is conjectured to have decidable type checking, and has an
implemented type-checker. Our new type theory, called guarded cubical type
theory, provides a computational interpretation of extensionality for guarded
recursive types. This further expands the foundations of CTT as a basis for
formalisation in mathematics and computer science. We present examples to
demonstrate the expressivity of our type theory, all of which have been checked
using a prototype type-checker implementation, and present semantics in a
presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201
A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic
The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension.
As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Omega. There are proofs which inhabit propositions, which are the terms of type Omega. The canonical propositions are those constructed from false by implication. Thirdly, there are paths which inhabit equations M =_A N, where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity.
We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
Partial Univalence in n-truncated Type Theory
It is well known that univalence is incompatible with uniqueness of identity
proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets
having non-trivial automorphisms as soon as they are not h-propositions.
A natural question is then whether univalence restricted to h-propositions is
compatible with UIP. We answer this affirmatively by constructing a model where
types are elements of a closed universe defined as a higher inductive type in
homotopy type theory. This universe has a path constructor for simultaneous
"partial" univalent completion, i.e., restricted to h-propositions.
More generally, we show that univalence restricted to -types is
consistent with the assumption that all types are -truncated. Moreover we
parametrize our construction by a suitably well-behaved container, to abstract
from a concrete choice of type formers for the universe.Comment: 21 pages, long version of paper accepted at LICS 202
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